light quanta

The equation explained

f(A_1, A_2 \ldots \alpha_1, \alpha_2 \ldots) = F_1(A_1) F_2(A_2) \ldots f_1(\alpha_1) \cdot f_2(\alpha_2) \ldots ,

In wordsThe joint probability density of all the amplitudes and all the phases is a product of separate factors, each depending on one amplitude or on one phase alone.OverviewRadiation is as disordered as possible when that density splits into separate factors, one for each amplitude and one for each phase. The value any one of them takes is then independent of the values of all the others.Full explanationThe footnote now states its condition: the radiation is the most disordered conceivable when the density of the preceding display is a product of separate factors, one for each amplitude

A_\nu

that depends on it alone (the paper's F with subscripts 1, 2 and so on) and one for each phase

\alpha_\nu

that depends on it alone (its f with the same subscripts). In the paper's words, the probability of a particular value of one amplitude or phase does not depend on the values the others have. It adds that the radiation of section 1 comes the closer to this condition the more each pair

A_\nu

\alpha_\nu

depends on emission and absorption by its own group of resonators. The footnote defines the condition; it does not show that the radiation in the enclosure meets it exactly.Every step

F_1(A_1)

For independent quantities the probability of a joint outcome is the product of the separate probabilities. So each amplitude gets a density of its own, depending on that amplitude alone; the paper writes these as F with the subscripts 1, 2 and so on.

f_1(\alpha_1)

The same for each phase, with a density of its own; the paper writes these as f with the same subscripts, in lower case.

f(A_1, A_2 \ldots \alpha_1, \alpha_2 \ldots) = F_1(A_1) F_2(A_2) \ldots f_1(\alpha_1) \cdot f_2(\alpha_2) \ldots

The condition as printed: the joint density is the product of all those separate factors, so no amplitude's density depends on what any other amplitude or phase happens to be.

d W = F_1(A_1) d A_1 \cdot F_2(A_2) d A_2 \ldots f_1(\alpha_1) d \alpha_1 \cdot f_2(\alpha_2) d \alpha_2 \ldots

Put into the preceding display, the probability splits into a product of separate probabilities, one for each amplitude and one for each phase. That is what the paper means by calling the radiation as disordered as conceivable.Historian's marginIn the sentence that follows, the plate of p. 135 prints 'A bez. x' where the sense requires α, the phases. The German face keeps the x as printed and marks it; the record is typo-x-for-alpha-p135 in docs/provenance/ap-17-132.md.AlsoOverviewFull explanationEvery stepHistorian's margin

Read it in the paper